Superconvergence points of Hermite spectral interpolation

TL;DR

This paper investigates the superconvergence properties of Hermite spectral interpolation and collocation methods, identifying points where derivatives converge faster, supported by numerical validation.

Contribution

It provides a detailed analysis of superconvergence points in Hermite spectral methods and extends the results to spectral collocation for differential equations.

Findings

Identification of superconvergence points for derivatives

Extension of analysis to spectral collocation methods

Numerical confirmation of superconvergence properties

Abstract

Hermite spectral method plays an important role in the numerical simulation of various partial differential equations (PDEs) on unbounded domains. In this work, we study the superconvergence properties of Hermite spectral interpolation, i.e., interpolation at the zeros of Hermite polynomials in the space spanned by Hermite functions. We identify the points at which the convergence rates of the first- and second-order derivatives of the interpolant converge faster. We further extend the analysis to the Hermite spectral collocation method in solving differential equations and identify the superconvergence points both for function and derivative values. Numerical examples are provided to confirm the analysis of superconvergence points.